Two breather solution of LPDE is derived based on two eigenvalues, \(\lambda _j\)  with \(j=1,2\) . They are expressed in terms of modulation frequencies of the two breathers, \(\kappa _j=2\sqrt{1+\lambda _j^2}\)  . The derivation technique named as Darboux transformation that we employ to derive the solution is described in [1]}. With \(n=2\) , in the general expression (REF ), the two-breather solution of Eq.(REF ) is given by:
\(\nonumber G_2&=&\frac{\kappa _1^2-\kappa _2^2}{\kappa _1 \kappa _2} \big \lbrace - \delta _2 \kappa _1^3\cos \left(t_{\text{s2}} \kappa _2\right) \cosh \left(V_{\text{H1}} x_{\text{s1}}\right)\\\nonumber &+&\kappa _2\cosh \left(V_{\text{H2}} x_{\text{s2}}\right) [ \delta _1 \kappa _2^2\cos \left(t_{\text{s1}} \kappa _1\right)\\\nonumber &+& \kappa _1\left(\kappa _1^2-\kappa _2^2\right)\cosh \left(V_{\text{H1}} x_{\text{s1}}\right) ]\big \rbrace \\\nonumber H_2&=&\frac{2 \left(\kappa _1^2-\kappa _2^2\right)}{\kappa _1 \kappa _2} \big \lbrace  \delta _2 \kappa _2 \sinh \left(V_{\text{H2}} x_{\text{s2}}\right)[\delta _1 \cos \left(t_{\text{s1}} \kappa _1\right) \\\nonumber &-& \kappa _1\cosh \left(V_{\text{H1}} x_{\text{s1}}\right)]+ \delta _1 \kappa _1\sinh \left(V_{\text{H1}} x_{\text{s1}}\right) \\\nonumber &\times &[\kappa _2 \cosh \left(V_{\text{H2}} x_{\text{s2}}\right)-\delta _2\cos \left(t_{\text{s2}} \kappa _2\right) ]\big \rbrace \\\nonumber D_1&=&\frac{1}{\kappa _1 \kappa _2} \big \lbrace \kappa _1 \cosh \left(V_{\text{H1}} x_{\text{s1}}\right)\lbrace 2 \delta _2 \left(\kappa _1^2-\kappa _2^2\right) \cos \left(t_{\text{s2}} \kappa _2\right)\\\nonumber &+&\kappa _2\cosh \left(V_{\text{H2}} x_{\text{s2}}\right)[\kappa _1^2 \left(\kappa _2^2-2\right)-2 \kappa _2^2] \rbrace \\\nonumber &+&2 \delta _1 \big [\delta _2 \left(\kappa _1^2+\kappa _2^2\right) \cos \left(t_{\text{s1}} \kappa _1\right) \cos \left(t_{\text{s2}} \kappa _2\right)\\\nonumber &+&\kappa _2 \big (2 \delta _2 \kappa _1[\sin \left(t_{\text{s1}} \kappa _1\right) \sin \left(t_{\text{s2}} \kappa _2\right)\\&+&\sinh \left(V_{\text{H1}} x_{\text{s1}}\right) \sinh \left(V_{\text{H2}} x_{\text{s2}}\right)]\\\nonumber &+&\left(\kappa _2^2-\kappa _1^2\right) \cos \left(t_{\text{s1}} \kappa _1\right) \cosh \left(V_{\text{H2}} x_{\text{s2}}\right)\big )\big ]\big \rbrace \) 
